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NonCommutativeGroebnerBasis [{poly1,poly2,…},vars,alg]

attempts to find a list of polynomials that form a reduced Gröbner basis for the set of polynomials polyi in variables vars over the non-commutative algebra alg.

NonCommutativeGroebnerBasis [{poly1,poly2,…},alg]

attempts to find a list of polynomials that form a reduced Gröbner basis for the set of polynomials polyi in the generators of the non-commutative algebra alg.

NonCommutativeGroebnerBasis [{poly1,poly2,…},alg,Left ]

finds a list of polynomials that form a reduced left Gröbner basis for the set of polynomials polyi in the generators of the algebra alg with commutation relations.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Free Algebras  
Quotient Algebras  
Algebras with Commutation Rules  
Options  
MaxIterations  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
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NonCommutativeGroebnerBasis [{poly1,poly2,…},vars,alg]

attempts to find a list of polynomials that form a reduced Gröbner basis for the set of polynomials polyi in variables vars over the non-commutative algebra alg.

NonCommutativeGroebnerBasis [{poly1,poly2,…},alg]

attempts to find a list of polynomials that form a reduced Gröbner basis for the set of polynomials polyi in the generators of the non-commutative algebra alg.

NonCommutativeGroebnerBasis [{poly1,poly2,…},alg,Left ]

finds a list of polynomials that form a reduced left Gröbner basis for the set of polynomials polyi in the generators of the algebra alg with commutation relations.

Details and Options

  • NonCommutativeGroebnerBasis is used to compute the reduced Gröbner basis of a set of polynomials over a non-commutative algebra.
  • alg can be a NonCommutativeAlgebra object or any valid NonCommutativeAlgebra specification. If the algebra argument is omitted, NonCommutativeAlgebra with the default property values is used.
  • vars needs to be specified when alg does not have generators specified ("Generators"Automatic ).
  • vars should be {vs1,…,vsk}, where vsi are disjoint lists of variables that include all non-commutative variables that appear in {poly1,poly2,…}. vsi=x, where x is a variable, is equivalent to vsi={x}.
  • vars, together with "VariableOrder" and "WordOrder" specifications in alg, determines a multi-graded monomial order.
  • "VariableOrder" can be either "Increasing" or "Decreasing". With the default "Increasing" setting, variables that appear earlier in variable lists are considered to be smaller. The elements of "CommutativeVariables" of the algebra alg are always considered smaller than all vars.
  • Monomials are ordered first on the number of occurrences of variables from variable lists vsi, starting with the list that contains the largest variables, then on the number of occurrences of "CommutativeVariables". If all numbers of occurrences are the same, monomials are ordered using "WordOrder", which can be one of "Lexicographic" (default), "ReverseLexicographic", "NegativeLexicographic" and "NegativeReverseLexicographic".
  • When alg has generators specified ("Generators"gens), {poly1,poly2,…} need to be polynomials in gens, and gens determines the monomial order.
  • A reduced Gröbner basis is a set gb of polynomials such that the leading monomial of each nonzero element of the ideal generated by {poly1,poly2,…} is divisible by the leading monomial of some element of gb, and no monomial of an element of gb is divisible by the leading monomial of another element of gb. For quotient algebras, that is, if alg has relations specified either through "Relations" or through "Structure", all polynomials are assumed to be in the standard form, that is, reduced modulo the Gröbner basis of the relations.
  • For a given monomial order, a reduced Gröbner basis is unique.
  • The notion of divisibility, and hence of Gröbner basis, differs depending on whether the algebra alg has commutation relations. For general non-commutative algebras (with "CommutationRelations"Automatic and "Structure"Automatic ), a monomial is divisible by a monomial if for some monomials and . For algebras with commutation relations, all algebra elements can be uniquely represented as linear combinations of standard monomials , where are the algebra generators, and the exponents denote repeated application of the algebra multiplication. In this case, a monomial is divisible by a monomial if for .
  • Left Gröbner basis computation is implemented only for algebras with commutation relations.
  • The following option can be given:
  • MaxIterations Automatic the maximum number of iterations to use
  • A finite Gröbner basis may not exist. The value of the MaxIterations option limits the number of algorithm iterations in which a new basis element is found. If the limit is reached, NonCommutativeGroebnerBasis issues a message and returns the basis elements found so far.
  • The Gröbner basis, and even finiteness of the Gröbner basis, in general depend on the ordering assigned to monomials.
  • For algebras with commutation relations, reduced Gröbner bases are always finite.

Examples

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Basic Examples  (4)

Compute the Gröbner basis over a free noncommutative algebra:

Wolfram Language code: NonCommutativeGroebnerBasis[{2 x**y**x + 3 x**x + 1, 3 y**x**x + 2 y**x - x**y}, {x, y}]

Prove that polynomials generate the whole algebra:

Wolfram Language code: NonCommutativeGroebnerBasis[{2 x**y**x + 3 x**x + 4 y + 5, 3 y**x**x + 2 y**x - x, y**x**y - 2x + 1}, {x, y}]

Compute the Gröbner basis over an algebra with commutation relations:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Generators" -> {{x, y}}, "CommutationRelations" -> {y**x - 2x**y + 3x + 4y + 5}|>]
Wolfram Language code: NonCommutativeGroebnerBasis[{ x**y**x**y + 2y**y**x**x - 3 y**x**x, 4 y**x**x**y + 5 y**x**y}, alg]

Compute the Gröbner basis of a left ideal in a Weyl algebra:

Wolfram Language code: alg = WeylAlgebra[{{x, y, z}, {dx, dy, dz}}]
Wolfram Language code: NonCommutativeGroebnerBasis[{dx + y**dz, dy - x}, alg, Left]

Scope  (11)

Free Algebras  (5)

Gröbner basis over the default algebra:

Wolfram Language code: NonCommutativeGroebnerBasis[{x**y - z, z**z - 2x}, {{x, y, z}}]

Gröbner basis over a free algebra with generators specified:

Wolfram Language code: alg = NonCommutativeAlgebra["Generators" -> {{x, y}}]
Wolfram Language code: NonCommutativeGroebnerBasis[{y**x**y - 2y + 1, y**y**y - 2x**x}, alg]

Finiteness of the Gröbner basis may depend on the monomial order:

Wolfram Language code: polys = {z**x + x**z**y, y**x - 1, x**y - 1};
Wolfram Language code: NonCommutativeGroebnerBasis[polys, {x, y, z}]//Short
Wolfram Language code: NonCommutativeGroebnerBasis[polys, {{x, y}, z}]

Gröbner basis over an algebra with commutative and scalar variables:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"ScalarVariables" -> {a, b}, "CommutativeVariables" -> {z, t}|>]; polys = {a t**x + b ^ 2 z**x**y + x**y + 5, 3 x**z + 5 x**x**t + a b};
Wolfram Language code: NonCommutativeGroebnerBasis[polys, {x, y}, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, %, {x, y}, alg]& /@ polys

Gröbner basis over an algebra with symbolic property names:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Multiplication" -> mult, "Addition" -> add, "Unity" -> one, "Zero" -> zero|>]; polys = {add[mult[x, y, x], z], add[mult[z, y], 2mult[x, y], 3]};
Wolfram Language code: NonCommutativeGroebnerBasis[polys, {x, y, z}, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, %, {x, y, z}, alg]& /@ polys

Quotient Algebras  (2)

Compute a Gröbner basis over a quotient algebra:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Relations" -> {y**y - x}|>]; polys = {y**x - z**x, z**z - 2x**y};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[polys, {{x, y, z}}, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, {{x, y, z}}, alg]& /@ polys

The Gröbner basis is a subset of the Gröbner basis of polys and the relations over the free algebra:

Wolfram Language code: fgb = NonCommutativeGroebnerBasis[Join[polys, {y**y - x}], {{x, y, z}}]

It consists of elements whose leading terms do not reduce modulo the Gröbner basis of the relations:

Wolfram Language code: lt = First[NonCommutativeMonomialList[#, {{x, y, z}}]]& /@ fgb
Wolfram Language code: rgb = NonCommutativeGroebnerBasis[{y**y - x}, {{x, y, z}}]
Wolfram Language code: rlt = NonCommutativePolynomialReduction[#, rgb, {{x, y, z}}]& /@ lt
Wolfram Language code: SymmetricDifference[First /@ Select[Transpose[{fgb, lt, rlt}], #[[2]] === #[[3]]&], gb]

Compute a Gröbner basis over an algebra given by generators and relations:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Generators" -> {{x, y, z}}, "Relations" -> {y**y - 2x**z}|>]; polys = {x**y**z - 1, z**z - 3y**y};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[polys, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, alg]& /@ polys

Algebras with Commutation Rules  (4)

Gröbner basis over an algebra with commutation relations specified:

Wolfram Language code: alg = NonCommutativeAlgebra[<|"Generators" -> {{x, y, z}}, "CommutationRelations" -> {y**x - 2x**y + x, z**y - 2y**z + z, z**x - 2x**z + y}|>]; polys = {x**y - 2z**z, x**y**z - 1};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[polys, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, alg]& /@ polys

Left Gröbner basis over an algebra with commutation relations specified:

Wolfram Language code: alg = WeylAlgebra[{{x, y, z}, {dx, dy, dz}}]; polys = {dx**dy - dz**z, x**y**z - x**z**dx};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[polys, alg, Left]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, alg]& /@ polys

Gröbner basis over a quotient algebra with commutation relations specified:

Wolfram Language code: alg = CliffordAlgebra[{{}, {x, y, z}}]; polys = {x**y**z - 1};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[polys, alg]

polys reduce to zero modulo the Gröbner basis:

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, alg]& /@ polys

Algebras with commutation relations have a different notion of divisibility than quotient algebras:

Wolfram Language code: rels = {Commutator[x, y], Commutator[x, z], Commutator[y, z]};

alg1 and alg2 are isomorphic, however divides in alg1, but not in alg2:

Wolfram Language code: alg1 = NonCommutativeAlgebra[<|"Generators" -> {x, y, z}, "CommutationRelations" -> rels|>];
Wolfram Language code: alg2 = NonCommutativeAlgebra[<|"Generators" -> {x, y, z}, "Relations" -> rels|>];

The Gröbner basis of over alg1 is finite, but over alg2 is infinite:

Wolfram Language code: NonCommutativeGroebnerBasis[{x**z}, alg1]
Wolfram Language code: NonCommutativeGroebnerBasis[{x**z}, alg2]

Options  (2)

MaxIterations  (2)

NonCommutativeGroebnerBasis attempts up to iterations for algebras without commutation rules:

Wolfram Language code: NonCommutativeGroebnerBasis[{b**b - b, b**a**b - a**b}, {a, b}]//AbsoluteTiming

Setting MaxIterations 10 makes NonCommutativeGroebnerBasis give up faster:

Wolfram Language code: NonCommutativeGroebnerBasis[{b**b - b, b**a**b - a**b}, {a, b}, MaxIterations -> 10]//AbsoluteTiming

A low value of MaxIterations may result in not finding a finite Gröbner basis even if one exists:

Wolfram Language code: NonCommutativeGroebnerBasis[{b**b - b, b**a**b - a**b, GeneralizedPower[NonCommutativeMultiply, a, 15] - a}, {a, b}, MaxIterations -> 10]
Wolfram Language code: NonCommutativeGroebnerBasis[{b**b - b, b**a**b - a**b, GeneralizedPower[NonCommutativeMultiply, a, 15] - a}, {a, b}]

Applications  (6)

Simplify an expression involving matrices and :

Wolfram Language code: expr = (Inverse[a] + Inverse[b]).a.Inverse[a + b].b

Find the inverses in the expression:

Wolfram Language code: inv = Union[Cases[expr, _Inverse, Infinity]]

Generate the relations satisfied by the inverses:

Wolfram Language code: rel = Join@@({#1.First[#1] - SymbolicIdentityArray[{n}], First[#1].#1 - SymbolicIdentityArray[{n}]}& /@ inv)

Pick a variable order that makes the most complicated variables largest:

Wolfram Language code: vars = Join[{a, b}, inv]

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Wolfram Language code: alg = NonCommutativeAlgebra[{Dot, n}];
Wolfram Language code: gb = NonCommutativeGroebnerBasis[rel, vars, alg]

Reducing the expression modulo the Gröbner basis shows that it is equal to the identity matrix:

Wolfram Language code: NonCommutativePolynomialReduction[expr, gb, vars, alg]

Use ArraySimplify to do the above simplification steps automatically:

Wolfram Language code: ArraySimplify[expr, Element[a | b, Matrices[{n, n}]]]

Prove the Woodbury matrix identity:

TemplateBox[{{(, {a, +, {u, ., c, ., v}}, )}}, Inverse]=TemplateBox[{a}, Inverse]-TemplateBox[{a}, Inverse].u.TemplateBox[{{(, {{v, ., TemplateBox[{a}, Inverse, SyntaxForm -> SuperscriptBox], ., u}, +, TemplateBox[{c}, Inverse, SyntaxForm -> SuperscriptBox]}, )}}, Inverse].v.TemplateBox[{a}, Inverse]

In the standard formulation, is an matrix, is a matrix, with , is an matrix, and is a matrix. However, by replacing , and with the block matrices , and , one may assume that all matrices belong to the algebra of matrices. It will be shown that the difference of the sides of the identity reduces to zero modulo the Gröbner basis of the ideal generated by relations implied by the properties of the matrix inverse.

Compute the difference of the sides of the Woodbury matrix identity:

Wolfram Language code: d = Subtract@@(Inverse[(a + u.c.v)] == Inverse[a] - Inverse[a].u.Inverse[(v.Inverse[a].u + Inverse[c])].v.Inverse[a])

Find the inverses in :

Wolfram Language code: inv = Union[Cases[d, _Inverse, Infinity]]

Generate the relations satisfied by the inverses:

Wolfram Language code: rel = Join@@({#1.First[#1] - SymbolicIdentityArray[{n}], First[#1].#1 - SymbolicIdentityArray[{n}]}& /@ inv)

Pick a variable order that makes the most complicated variables largest:

Wolfram Language code: vars = Join[{a, c, u, v}, inv]

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Wolfram Language code: alg = NonCommutativeAlgebra[{Dot, n}];
Wolfram Language code: gb = NonCommutativeGroebnerBasis[rel, vars, alg]

Reduce modulo the Gröbner basis. The result is a zero matrix, which proves the identity:

Wolfram Language code: NonCommutativePolynomialReduction[d, gb, vars, alg]

Use ArraySimplify to do the above simplification steps automatically:

Wolfram Language code: ArraySimplify[d, Element[a | c | u | v, Matrices[{n, n}]]]

Construct a finitely presented group. The dicyclic group is given by generators and relations . The group algebra of is given by four generators:

Wolfram Language code: gens = {a, x, inv[a], inv[x]};

The generators satisfy the following relations:

Wolfram Language code: rel = {GeneralizedPower[NonCommutativeMultiply, a, 6] - 1, GeneralizedPower[NonCommutativeMultiply, x, 2] - GeneralizedPower[NonCommutativeMultiply, a, 3], inv[x]**a**x - inv[a], a**inv[a] - 1, inv[a]**a - 1, x**inv[x] - 1, inv[x]**x - 1};

Compute the Gröbner basis of the ideal generated by the relations:

Wolfram Language code: vars = {{a, x}, {inv[a], inv[x]}};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[rel, vars]

Note that reducing an arbitrary monomial modulo the Gröbner basis gives a monomial that does not contain and . This shows that reducing an arbitrary monomial in and of total degree yields a monomial of a total degree at most :

Wolfram Language code: NonCommutativePolynomialReduction[#, gb, vars]& /@ NonCommutativeMultiply@@@Tuples[{a, x}, 5]

Hence, all elements of the group can be represented as reduced monomials of degree at most :

Wolfram Language code: gg = Union[NonCommutativePolynomialReduction[#, gb, vars]& /@ Join@@Table[NonCommutativeMultiply@@@Tuples[{a, x}, i], {i, 0, 4}]]

It has been proven that is a finite group of order :

Wolfram Language code: Length[gg]

Test left ideal membership in a Clifford algebra:

Wolfram Language code: alg = CliffordAlgebra[{{x, y}, {u, v}}]

Compute the Gröbner basis of the left ideal :

Wolfram Language code: gb = NonCommutativeGroebnerBasis[{x**y - u**v}, alg, Left]

A polynomial belongs to the ideal if it reduces to :

Wolfram Language code: NonCommutativePolynomialReduction[x**u**v, gb, alg]
Wolfram Language code: NonCommutativePolynomialReduction[x**u**v**y + 1, gb, alg]

Solve an overdetermined system of homogenous linear ODEs with polynomial coefficients:

Wolfram Language code: eqns = {-y[x] - Derivative[1][y][x] - 2 Derivative[2][y][x] + Derivative[3][y][x] - x Derivative[3][y][x] + 3 Derivative[4][y][x] + x Derivative[5][y][x] == 0, 2 x^2 y[x] - 2 x^2 Derivative[2][y][x] - Derivative[3][y][x] + Derivative[5][y][x] == 0, 3 y[x] - 6 Derivative[2][y][x] - x Derivative[3][y][x] + 3 Derivative[4][y][x] + x Derivative[5][y][x] == 0};

Convert equations to Weyl algebra elements, with representing differentiation with respect to :

Wolfram Language code: toWeyl[a_ == b_] := toWeyl[a - b] toWeyl[a : (_List | _Plus)] := toWeyl /@ a toWeyl[c_ ? NumericQ a_] := c toWeyl[a] toWeyl[a_] := a /. {Derivative[n_][y][x] -> z ^ n, y[x] -> 1} /. Times -> NonCommutativeMultiply /. {Power[x, n_.] -> GeneralizedPower[NonCommutativeMultiply, x, n], Power[z, n_.] -> GeneralizedPower[NonCommutativeMultiply, dx, n]}
Wolfram Language code: wpolys = toWeyl[eqns]

Compute the Gröbner basis of the left ideal generated by the Weyl algebra elements:

Wolfram Language code: wa = WeylAlgebra[{{x}, {dx}}]
Wolfram Language code: gb = NonCommutativeGroebnerBasis[wpolys, wa, Left]

Find the differential equation corresponding to the Gröbner basis:

Wolfram Language code: toDiff[f_Plus] := toDiff /@ f toDiff[f_] := If[FreeQ[f, dx], (f /. NonCommutativeMultiply -> Times) y[x], (f /. NonCommutativeMultiply -> Times) /. Power[dx, n_.] :> D[y[x], {x, n}]] toEqn[f_List] := toEqn /@ f toEqn[f_] := toDiff[f] == 0
Wolfram Language code: eqn = toEqn[gb]

Solve the differential equation:

Wolfram Language code: sol = DSolve[eqn, y, x]

Verify that the solution satisfies the original equations:

Wolfram Language code: eqns /. sol

Solve an overdetermined system of homogenous linear PDEs with polynomial coefficients:

Wolfram Language code: eqns = {u[x, y] + 8 y u[x, y] + 8 x y u[x, y] + u^(0, 1)[x, y] + 2 x u^(0, 1)[x, y] + u^(1, 0)[x, y] + x^2 u^(1, 0)[x, y] + 4 y u^(1, 0)[x, y] + 4 x^2 y u^(1, 0)[x, y] == 0, -3 u^(1, 0)[x, y] - 6 x u^(1, 0)[x, y] + 4 y u^(1, 0)[x, y] + 2 u^(1, 1)[x, y] + 2 x u^(1, 1)[x, y] - 3 u^(2, 0)[x, y] - 3 x^2 u^(2, 0)[x, y] + 4 y u^(2, 0)[x, y] + 2 u^(2, 1)[x, y] + x^2 u^(2, 1)[x, y] == 0, y u[x, y] + 4 x y u[x, y] + u^(0, 1)[x, y] + x u^(0, 1)[x, y] - 3 y u^(1, 0)[x, y] + x^2 y u^(1, 0)[x, y] + x^2 u^(1, 1)[x, y] == 0};

Convert equations to Weyl algebra elements, with representing differentiation with respect to :

Wolfram Language code: toWeyl[a_ == b_] := toWeyl[a - b] toWeyl[a : (_List | _Plus)] := toWeyl /@ a toWeyl[c_ ? NumericQ a_] := c toWeyl[a] toWeyl[a_] := a /. {Derivative[n_, m_][u][x, y] -> zx ^ n zy ^ m, u[x, y] -> 1} /. Times -> NonCommutativeMultiply /. {Power[x, n_.] -> GeneralizedPower[NonCommutativeMultiply, x, n], Power[zx, n_.] -> GeneralizedPower[NonCommutativeMultiply, dx, n], Power[zy, n_.] -> GeneralizedPower[NonCommutativeMultiply, dy, n]}
Wolfram Language code: wpolys = toWeyl[eqns]

Compute the Gröbner basis of the left ideal generated by the Weyl algebra elements:

Wolfram Language code: wa = WeylAlgebra[{{x, y}, {dx, dy}}]
Wolfram Language code: gb = NonCommutativeGroebnerBasis[wpolys, wa, Left]

Find the differential equations corresponding to the Gröbner basis:

Wolfram Language code: toDiff[f_Plus] := toDiff /@ f toDiff[f_] := If[FreeQ[f, dx | dy], (f /. NonCommutativeMultiply -> Times) u[x, y], (f /. NonCommutativeMultiply -> Times) /. {Power[dx, n_.]Power[dy, m_.] :> D[u[x, y], {x, n}, {y, m}], Power[dx, n_.] :> D[u[x, y], {x, n}], Power[dy, n_.] :> D[u[x, y], {y, n}]}] toEqn[f_List] := toEqn /@ f toEqn[f_] := toDiff[f] == 0
Wolfram Language code: eqn = toEqn[gb]

Solve the differential equations:

Wolfram Language code: sol = DSolve[eqn, u, {x, y}]

Verify that the solution satisfies the original equations:

Wolfram Language code: Simplify[eqns /. sol]

Properties & Relations  (2)

A Gröbner basis generates the same ideal as the input polynomials:

Wolfram Language code: {p, q} = {y**y - 2y**x**y, y**x - 2 x**x};
Wolfram Language code: gb = NonCommutativeGroebnerBasis[{p, q}, {{x, y}}]

Use NonCommutativePolynomialReduction to show that p is in the ideal generated by gb:

Wolfram Language code: NonCommutativePolynomialReduction[p, gb, {{x, y}}]

Use NonCommutativePolynomialReduce to represent p as a combination of basis elements:

Wolfram Language code: {fs, r} = NonCommutativePolynomialReduce[p, gb, {{x, y}}]
Wolfram Language code: Plus@@MapThread[Construct, {fs, gb}]
Wolfram Language code: NonCommutativeExpand[% - p]

Use GroebnerBasis to compute a Gröbner basis of commutative polynomials:

Wolfram Language code: GroebnerBasis[{x * x - 2x * y * x, x * y - 2 y * y}, {x, y}]

Tech Notes

Wolfram Research (2025), NonCommutativeGroebnerBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html (updated 2026).

Text

Wolfram Research (2025), NonCommutativeGroebnerBasis, Wolfram Language function, https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html (updated 2026).

CMS

Wolfram Language. 2025. "NonCommutativeGroebnerBasis." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2026. https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html.

APA

Wolfram Language. (2025). NonCommutativeGroebnerBasis. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html

BibTeX

@misc{reference.wolfram_2026_noncommutativegroebnerbasis, author="Wolfram Research", title="{NonCommutativeGroebnerBasis}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html}", note=[Accessed: 03-July-2026]}

BibLaTeX

@online{reference.wolfram_2026_noncommutativegroebnerbasis, organization={Wolfram Research}, title={NonCommutativeGroebnerBasis}, year={2026}, url={https://reference.wolfram.com/language/ref/NonCommutativeGroebnerBasis.html}, note=[Accessed: 03-July-2026]}
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